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Positive Maps , [Online http://www.math.tamu.edu/~pskoufra/OANotes-CompletelyPositiveMaps.pdf ]. [12] J. Pečarić, T. Furuta, J. Mićić Hot and Y. Seo, Mond-Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space , Element, Zagreb, 2005. [13] S.-H. Wang and X.-M. Liu, Hermite-Hadamard type inequalities for operator s-preinvex functions. J. Nonlinear Sci. Appl. 8 2015, no. 6, 1070–1081.

topological spaces were defined in the context of C * {C^{*}} -algebras. In [ 9 , 10 ], completely positive maps of order zero with finite-dimensional domains were used to define noncommutative versions of topological covering dimension. A completely positive map between two C * {C^{*}} -algebras has order zero if it preserves orthogonality. Wolff [ 13 ] described the structure of bounded linear self-adjoint maps with unital domains that preserve orthogonality. He showed that any such map is the compression of a Jordan * {*} -morphism with a self-adjoint element

Chapter Three Completely Positive Maps For several reasons a special class of positive maps, called completely positive maps, is especially important. In Section 3.1 we study the basic properties of this class of maps. In Section 3.3 we derive some Schwarz type inequalities for this class; these are not always true for all positive maps. In Sections 3.4 and 3.5 we use general results on completely positive maps to study some important problems for matrix norms. Let Mm(Mn) be the space of m×m block matrices [[Aij ]] whose i, j entry is an element of Mn = Mn

Forum Math. 3 (1991), 389-400 Forum Mathematicum © de Gruyter 1991 A Note on Quasicontinuous Kernels Representing Quasi- Linear Positive Maps Sergio Albeverio*'**'* and Zhi-Ming Ma**'*** (Communicated by Masatoshi Fukushima) Abstract. We consider topological spaces with an outer capacity. We construct quasicontinuous (relative to the given capacity) kernels representing quasi-linear positive maps. The application of this result to Lusin and Radon spaces yields a construction of right Markov processes associated with Dirichlet forms without assumption of

## Abstract

In this paper, we prove a covariant version of the Stinespring theorem for Hilbert C*-modules. Also, we show that there is a bijective correspondence between operator valued completely positive maps, (u′, u)-covariant with respect to the dynamical system (G, η, X) on Hilbert C*-modules and (u′, u)-covariant operator valued completely positive maps on the crossed product G ×η X of X by η.

## Abstract

In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

presented. In Section 3 , we prove our main result, the classification of the Cauchy transforms in this non-commutative setting. In Section 4 , we prove an analogous characterization of the linearizing transforms associated to ℬ {\mathcal{B}} -valued distributions. In Section 5 , we derive some of the many consequences of this result, including Nevanlinna type representations for certain classes of non-commutative functions and defining semigroups of completely positive maps associated to each infinitely divisible distribution. 2 Preliminaries 2.1 Vector

## Noise effects in the quantum search algorithm from the viewpoint of computational complexity

We analyse the resilience of the quantum search algorithm in the presence of quantum noise modelled as trace preserving completely positive maps. We study the influence of noise on the computational complexity of the quantum search algorithm. We show that it is only for small amounts of noise that the quantum search algorithm is still more efficient than any classical algorithm.

## Natural Quantum Operational Semantics with Predicates

A general definition of a quantum predicate and quantum labelled transition systems for finite quantum computation systems is presented. The notion of a quantum predicate as a positive operator-valued measure is developed. The main results of this paper are a theorem about the existence of generalised predicates for quantum programs defined as completely positive maps and a theorem about the existence of a GSOS format for quantum labelled transition systems. The first theorem is a slight generalisation of D'Hondt and Panagaden's theorem about the quantum weakest precondition in terms of discrete support positive operator-valued measures.